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EXACT  ASTRONOMY. 


A  Dynamical  Solution  of  the 
Fundamental  Problems 

OF 

Mathematical  Astronomy. 


BY 

MYRON  HUTCHINSON. 


^   i-ii-U;i  J'  Y'' 


Exact  Astronomy. 


A    DYNAMICAL   SOLUTION   OF   THE 
FUNDAMENTAL   PROBLExMS 

OF 

Mathematical  Astronomy. 


MYRON  HUTCHINSON. 


SAN  FRANCISCO: 

A.  J    Leary,  Printer,  402-408  Sansome  St. 

1889. 


Entered  according  to  Act  of  Congress,  in  the  year  iccg. 

By  MYRON   HUTCHINSON, 
in  the  ofiice  of  the  Librarian  of  Congress,  at  Washingtoi 


TO 

WALTER  WILLIAM  PALMER, 

In  recognition  of  the  stimulant  to  persistence  wrought  by  his  kindness 
and  inteUigent  appreciation,  this  work  is  respectfully  dedicated 

By  his  friend, 

The  Author. 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/exactastronomydyOOhutcrich 


("" 


INTRODUCTORY. 


Distance  and  Dimensions  of  the  Sun. 

From  the  "  Sun,"  by    Professor  C.  A.  Young. 


The  problem  of  fiading  the  distance  of  the  sun  is 
one  of  the  most  important  and  difficult  presented  by  as- 
tronomy. Its  importance  lies  in  this,  that  this  distance — 
the  radius  of  the  earth's  orbit — is  the  base  line  by  means  of 
which  we  measure  every  other  celestial  distance,  excepting 
only  that  of  the  moon  ;  so  that  error  in  this  base  propagates 
itself  in  all  directions,  through  all  space,  affecting  with  a 
corresponding  proportion  of  falsehood  every  measured  line, 
the  distance  of  every  star,  the  radius  of  every  orbit,  the 
diameter  of  every  planet. 

Our  estimates  of  the  masses  of  the  heavenly  bodies 
also  depends  upon  a  knowledge  of  the  sun's  distance  from 
the  earth.  The  quantity  of  matter  in  a  star  or  planet  is  de- 
termined by  calculations  whose  fundamental  data  include 
the  distance  between  the  investigated  body,  and  some 
other  body  whose  motion  is  controlled  or  modified  by  it;  so 
that  any  error  in  it  involves  a  more  than  three-fold  error  in 
the  resulting  mass.  An  uncertainty  of  one  per  cent,  in  the 
sun's  distance  implies  an  uncertainty  of  more  than  three 
per  cent,  in  every  celestial  mass  and   every  cosmical  force. 

Error    in    this    fundamental  element  propagates   itself 
in  time  also,  as  well  as  in  space  and    mass.      That  is  to  say, 


6  INTRODUCTORY. 

our  calculations  of  the  mutual  effects  of  the  planets  upon 
each  other's  motions  depend  upon  an  accurate  knowledore  of 
their  masses  and  distances.  By  these  calculations,  were 
our  data  perfect,  we  could  predict  for  all  futurity,  or  repro- 
duce for  any  given  epoch  of  the  past,  the  configurations  of 
the  planets  and  the  conditions  of  their  orb  itsand  many  in- 
teresting problems  in  geology  and  natural  history  seem  to 
require  for  their  solution  just  such  determinations  of  the 
form  and  position  of  the  earth's  orbit  in  by-gone  ages. 

Now,  the  slightest  error  in  the  data,  though  hardly  af- 
fecting the  result  for  epochs  near  the  present,  leads  to 
uncertainty,  which  accumulates  with  extreme  rapidity  in  the 
lapse  of  time  ;  so  that  even  the  present  uncertainty  of  the 
sun's  distance,  small  as  it  is,  renders  precarious  all  conclu- 
sions from  such  computations  when  the  period  is  extended 
more  than  a  few  hundred  thousand  years  from  the  pres- 
ent time.  If,  for  instance,  we  should  find  as  the  result  of 
calculation  with  the  received  data,  that  two  millions  of 
years  ago  the  eccentricity  of  the  earth's  orbit  was  at  a 
maximum,  and  the  perihelion  so  placed  that  the  sun  was 
nearest  during  the  northern  winter  (a  condition  of  affairs 
which  is  thought  would  produce  a  glacial  epoch  in  the 
southern  hemisphere),  it  might  easily  happen  that  our  re- 
sults would  be  exactly  contrary  to  the  truth,  and  that  the 
state  of  affairs  indicated  did  not  occur  within  half  a  million 
years  of  the  specified  date  ;  and  all  because  in  our  calcu- 
lation the  sun's  distance,  or  solar  parallax  by  which  it  is 
measured,  was  assumed  half  of  one  per  cent,  too  great  or 
too  small.  In  fact,  this  solar  parallax  enters  into  almost 
every  kind  of  astronomical  computations,  from  those  which 
deal  with  stellar  systems  and  the  constitution  of  the  uni- 
verse., to  those  which  have  for  their  object  nothing  higher 
than  the  prediction  of  the  moon's  place  as  a  means  of  find- 
ing the  longitude  at  sea.     Of  course,  it  hardly  need  be  said 


INTRODUCTORY,  7 

that  its  determination  is  the  first  step  to  any  knowledge  of 
the  dimensions  and  constitution  of  the  sun  itself. 

This  parallax  of  the  sun  is  simply  the  angular  semi- 
diameter  of  the  earth  as  seen  from  the  sun  ;  or,  it  may  be 
defined  in  another  way  as  the  angle  between  the  direc- 
tion of  the  sun  ideally  observed  from  the  center  of  the 
earth,  and  its  actual  direction  as  seen  fr  )m  a  station  where 
it  is  just  rising  above  the  horizon. 

We  know  with  great  accuracy  the  dimensions  of  the 
earth.  Its  mean  equatorial  radius  according  to  the  latest 
and  most  reliable  determination  (agreeing,  however,  very 
closely  with  previous  ones,)  is  3962.720  English  miles,  and 
the  error  can  hardly  amount  to  more  than  0.0000 1  of  the 
whole — perhaps  200  feet  one  way  or  the  other.  Accord- 
ingly, if  we  know  how  large  the  earth  looks  from  any  point, 
or,  to  speak  technically,  if  we  know  the  parallax  of  the  point, 
its  distance  can  at  once  be  found  by  a  very  easy  calculation  ; 
it  equals  simply  (206265  x  the  radius  of  the  earth)-^-(the 
parallax  in  seconds  of  arc). 

Now,  in  the  case  of  the  sun  it  is  very  difficult  to  find 
the  parallax  with  sufficient  precision,  on  account  of  its  small- 
ness — it  is  less  than  9'',  almost  certainly  between  8". 8  and 
8".9.  But  this  tenth  of  a  second  of  doubtfulness  is  more 
than.o.oi  of  the  whole,  although  it  is  no  more  than  the  angle 
subtended  by  a  single  hair  at  a  distance  of  nearly  800  feet. 
If  we  call  the  parallax  8'''.86,  which  is  probably  very  near 
the  truth,  the  distance  of  the  sun  will  come  out  92,254,000 
miles,  while  a  variation  of  ^V  of  a  second  either  way  will 
change  it  nearly  half  a  million   miles. 

When  a  surveyor  has  to  find  the  distance  of  an  inaccess- 
ible object  he  lays  off  a  convenient  base  line,  and  from  its 
extremities  observes  the  directions  of  the  object,  considering 
himself  very  unfortunate  if  he  cannot  get  a  base  whose 
length  is  at  least  iV  of  the  distance  to  be  measured.      But  the 


8  INTRODUCTORY. 

whole  diameter  of  the  earth  is  less  than  7^  of  the  distance 
of  the  sun,  and  the  astronomer  is  in  the  predicament  of  a 
surveyor,  who,  having  to  measure  the  distance  of  an  object 
ten  miles  off,  finds  himself  restricted  to  a  base  ot  less  than 
five  feet,  and  herein  lies  the  difficulty  of  the  problem. 

Of  course,  it  would  be  hopeless  to  attempt  this  problem 
by  direct  observations,  such  as  answer  perfectly  in  the  case 
of  the  moon,  whose  distance  is  only  thirty  times  the  earth's 
diameter.  In  her  case,  observations  taken  from  stations 
widely  separated  in  latitude,  like  Berlin  and  the  Cape  of 
Good  Hope,  or  Washington  and  Santiago,  determine  her 
parallax  and  distance  with  very  satisfactory  precision ;  (very 
unsatisfactory,  rather,  since  the  error  is  very  nearly  half  of 
one  per  cent. — Author)  but  if  observations  of  the  same 
accuracy  could  be  made  upon  the  sun,  (which  is  not  the  case, 
since  its  heat  disturbs  the  adjustments  of  an  instrument)  they 
would  only  show  the  parallax  to  be  somewhere  between  8^' 
and  10'', — its  distance  between  126,000,000  and  82,000,000 
miles. 

Astronomers,  therefore,  have  been  driven  to  employ 
indirect  methods,  based  on  various  principles. 


On  the  Method  of  Zadig. 

From  a  Lecture  by  Huxley. 

It  is  a  usual  and  a  commendable  practice  to  preface  the 
discussion  of  the  views  of  a  philosophic  thinker  by  some 
account  of  the  man. and  of  the  circumstances  which  shaped 
his  life  and  colored  his  way  of  looking  at  things ;  but, 
though  Zadig  is  cited  in  one  of  the  most  important  chapters 
of  Cuvier's  greatest  work,  little  is  known  about  him,  and 
that  little  might,  perhaps,  be  better  authenticated  than  it  is. 


INTRODUCTORY.  9 

It  is  said  that  he  lived  at  Babylon  in  the  time  of  King 
Moabdar  ;  but  the  name  of  Moabdar  does  not  appear  in  the 
list  of  Babylonian  sovereigns  brought  to  light  by  the  patience 
and  the  industry  of  the  decipherers  of  cuneiform  inscriptions 
in  these  later  years ;  nor  Indeed  am  I  aware  that  there  is  any 
other  authority  for  his  existence  than  that  of  the  biogra- 
pher of  Zadig,  one  Arouet  de  Voltaire,  among  whose  most 
conspicuous  merits  strict  historical  accuracy  is  perhaps  hardly 
to  be  reckoned. 

Happily  Zadig  is  in  the  position  of  a  great  many  other 
philosophers.  What  he  was  like  when  he  was  in  the  flesh, 
indeed  whether  he  existed  at  all,  are  matters  of  no  great 
consequence.  What  we  care  about  in  a  light  is  that  it  shows 
the  way,  not  whether  it  is  lamp  or  candle,  tallow  or  wax. 
Our  only  real  interest  in  Zadig,  lies  in  the  conceptions  of 
which  he  is  the  putative  father;  and  his  biographer  has  stated 
these  with  so  much  clearness  and  vivacious  illustration, 
that  we  need  hardly  feel  a  pang  even  if  critical  research 
should  prove  King  Moabdar  and  the  rest  of  the  story  to  be 
unhistorical,  and  reduce  Zadig  himself  to  the  shadowy  con- 
dition of  a  solar  myth.  Voltaire  tells  us  that,  disenchanted 
with  life  by  sundry  domestic  misadventures,  Zadig  withdrew 
from  the  turmoil  of  Babylon  to  a  secluded  retreat  on  the 
banks  of  the  Euphrates,  where  he  beguiled  his  solitude  by 
the  study  of  nature.  The  manifold  wonders  of  the  world  of 
life  had  a  peculiar  attraction  for  the  lonely  student  ;  inces- 
sant and  patient  observation  of  the  planets  and  animals 
about  him  sharpened  his  naturally  good  powers  of  observa- 
tion and  of  reasoning;  until  it  length  he  acquired  a  saga- 
city which  enabled  him  to  perceive  endless  minute  differ- 
ences among  objects  which,  to  the  untutored  eye,  appear 
absolutely  alike. 

It  might  have  been  expected  that  this  enlargement  of  the 
powers  of  the  mind  and  of  its  store  of  natural  knowledge 


1^: 


lO  INTRODUCTORY. 

could  tend  to  nothing  but  the  Increase  of  a  man's  own 
welfare  and  the  good  of  his  fellow  men.  But  Zadig  was 
fated  to  experience  the  vanity  of  such  expectations. 

One  day  walking  near  a  little  wood,  he  saw,  hastening 
that  way,  one  of  the  queen's  chief  eunuchs,  followed  by  a 
troop  of  officials  who  appeared  to  be  in  the  greatest  anxiety, 
running  hither  and  thither  like  men  distraught,  In  search  of 
some  lost  treasure. 

''Young  man,"  cried  the  eunuch,  "have  you  seen  the 
queen's  dog  ?"  Zadig  answered  modestly,  *' A  bitch,  I  think, 
not  a  dog."  "  Quite  right,"  replied  the  eunuch  ;  and  Zadig 
continued  :  "A  very  small  spaniel  who  has  lately  had  pup- 
pies ;  she  limps  with  the  left  fore  leg,  and  has  very  long 
ears."  "  Ah!  you  have  seen  her,  then  ?''  said  the  breathless 
eunuch.  "  No,"  answered  Zadig,  "  I  have  not  seen  her  ; 
and  I  really  was  not  aware  that  the  queen  possessed  a 
spaniel." 

By  an  odd  coincidence,  at  the  very  same  time,  the  hand- 
somest horse  in  the  king's  stables  broke  away  from  his  groom 
in  the  Babylonian  plains.  The  grand  huntsman  and  all  his 
staff  were  seeking  the  horse  with  as  much  anxiety  as  the 
eunuch  and  his  people,  the  spaniel;  and  the  grand  huntsman 
asked  Zadig  If  he  had  not  seen  the  king's  horse  go  that 
way.  '' A  first-rate  galloper,  small  hoofed,  five  feet  high; 
tail  three  feet  and  a  half  long ;  cheek  pieces  of  the  bit  of 
twenty-three  carat  gold  ;  shoes  silver,"  said  Zadig. 

•' Which  way  did  he  go?  Where  Is  he.^"  cried  the 
grand  huntsman. 

''  I  have  not  seen  anything  of  the  horse,  and  I  never 
heard  of  him  before,"   replied  Zadig. 

The  grand  huntsman  and  the  chief  eunuch  made  sure 
that  Zadig  had  stolen  both  the  king's  horse  and  the  queen's 
spaniel;  so  they  haled  him  before  the  high  Court  of  Dester- 
ham,  which  at  once  condemned  him  to  the  knout  and  trans- 


INTRODUCTORY.  I  I 

portation  for  life  to  Siberia.  But  the  sentence  was  hardly 
pronounced  when  the  lost  horse  and  spaniel  were  found.  So 
the  judges  were  under  the  painful  necessity  of  reconsidering 
their  decision,  but  they  fined  Zadig  four  hundred  ounces  of 
gold  for  saying  that  he  had  seen  that  which  he  had  not  seen. 

The  first  thing  was  to  pay  the  fine  ;  afterwards  Zadig 
was  permitted  to  open  his  defense  to  the  Court,  which  he 
did  in  the  following  terms  : 

*'  Stars  of  justice,  abysses  of  knowledge,  mirrors  of  truth, 
whose  gravity  is  as  that  of  lead,  whose  inflexibility  is  as  that 
of  iron,  who  rival  the  diamond  in  clearness,  and  possess  no 
little  affinity  with  gold:—  Since  I  am  permitted  to  address  your 
august  assembly,  I  swear  by  Ormuzd  that  I  have  never  seen 
the  respectable  lady  dog  of  the  queen  nor  beheld  the  sacro- 
sanct horse  of  the  king  of  kings. 

''This  is  what  happened  :  I  was  taking  a  walk  toward 
the  little  wood  near  which  I  subsequently  had  the  honor  to 
meet  the  venerable  chief  eunuch  and  the  most  illustrious 
grand  huntsman.  I  noticed  the  track  of  an  animal  in  the 
sand,  and  it  was  easy  to  see  that  it  was  that  of  a  small  dog. 
Long,  faint  streaks  upon  the  little  elevations  of  sand  between 
foot  marks  convinced  me  that  it  was  a  she  dog  with  pend- 
ent dugs,  showing  that  she  must  have  had  puppies  not 
many  days  since.  Other  scrapings  of  the  sand,  which  al- 
ways lay  close  to  the  marks  of  the  fore  paws,  indicated  that 
she  had  very  long  ears ;  and  as  the  imprint  of  one  foot  was 
always  fainter  than  those  of  the  other  three,  I  judged  that 
the  lady  dog  of  our  august  queen  was,  if  I  may  venture  to 
say  so,  a  little  lame. 

"With  respect  to  the  horse  of  the  king  of  kings,  permit 
me  to  observe  that,  wandering  through  the  paths  which  trav- 
erse the  wood,  I  noticed  the  marks  of  horseshoes.  They 
were  all  equidistant.  '  Ah,'  said  I,  'this  Is  a  famous  gallop- 
er.'     In  a  narrow  alley,  only  seven  feet  wide,  the  dust  upon 


1 2  INTRODUCTORY. 

the  trunks  of  the  trees  was  a  little  disturbed  at  three  feet 
and  a  half  from  the  middle  of  the  path.  'This  horse/  said 
I  to  myself,  '  had  a  tail  three  feet  and  a  half  long,  and,  lash- 
ing it  from  one  side  to  the  other,  he  has  swept  away  the 
dust.'  Branches  of  trees  met  overhead  at  the  height  of  five 
feet,  and  under  them  I  saw  newly  fallen  leaves  ;  so  I  knew 
the  horse  had  brushed  some  of  the  branches  and  was  there- 
fore five  feet  high.  As  to  his  bit,  it  must  have  been  of 
twenty-three  carat  gold,  for  he  had  rubbed  it  aganist  a  stone 
which  turned  out  to  be  a  touchstone,  with  the  properties  of 
which  I  am  familiar  by  experiment.  Lastly,  by  the  marks 
which  his  shoes  left  upon  pebbles  of  another  kind,  I  was  led 
to  think  that  his  shoes  were  of  fine  silver." 

All  the  judges  admired  Zadig's  profound  and  subtle 
discernment  ;  and  the  fame  of  it  reached  even  the  king  and 
the  queen.  From  the  ante-rooms  to  the  presence-chamber, 
Zadig's  name  was  in  everybody's  mouth  ;  and,  although 
many  of  the  magi  were  of  the  opinion  that  he  ought  to  be 
burned  as  a  sorcerer,  the  king  commanded  that  the  four 
hundred  ounces  of  gold  which  he  had  been  fined  should  be 
restored  to  him.  So  the  officers  of  the  court  went  in  state 
with  the  four  hundred  ounces ;  only  they  retained  three  hun- 
dred and  ninety-eight  for  legal  expenses,  and  their  servants 
expected   fees. 

Those  who  are  interested  in  learning  more  of  the 
fateful  history  of  Zadig  must  turn  to  the  original ;  we 
are  dealing  with  him  only  as  a  philosopher,  and  this  brief 
excerpt  suffices  for  the  exemplification  of  the  nature  of 
his  conclusions  and  of  the  method  by  which  he  arrived  at 
them. 


PREFACE. 


These  few  pages  effect  the  exhaustive  solution  of  the 
fundamental  problems  of  mathematical  astronomy,  by  the 
application,  after  the  manner  of  Zadig,  of  the  elementary 
mathematics  only  to  accurately  known  data. 

The  remark  In  the  Introduction  to  Herschel's  **  Out- 
lines of  Astronomy"  that  (together  with  the  story  of  Zadig) 
suggested  to  my  mind  the  mode  of  attacking  the  great  prob- 
lem of  the  sun's  distance,  best  expresses  the  thought  I  wish 
to  enunciate, — ''  the  pearls  of  analytical  research  are  invar- 
iably strung  on  the  central  thread  of  common  sense." 

The  substitution  of  dynamical  for  statical  treatment  of 
the  problem,  has  revolutionized  the  oldest  of  the  sciences, 
and  dissipated  the  Intellectual  fog  in  which  abstruse  meth- 
ods and  deference  to  authority  have  so   long  enshrouded  it. 

Human  nature  remains  the  same  through  all  ages; 
and  my  experience  of  the  reception  accorded  truths  newly 
discovered  by  a  layman,  by  the  guild  of  scientists,  demon- 
strates that  Voltaire's  caustic  satire  is  as  applicable  to  the 
scientific  pretension,  arrogance  and  dishonesty  of  to-day  as 
to  that  of  his  own  time. 

MYRON   HUTCHINSON, 

San  Francisco,  October,  1889. 


EXACT  ASTRONOMY. 


A  Dynamical  Resolution  of  the  Solar  Parallax. 

The  conception  of  the  solar  horizontal  parallax  postu- 
lates a  circular  orbit  described  with  the  radius  R,  equal  to  the 
mean  distance  between  the  centers  of  terrestrial  and  lunar 
revolution,  measured  by  an  arc  x  of  the  said  great  circle  ; 
equal  in  length  to  the  earth's  equatorial  radius.  If  x  be 
taken  in  seconds,  so  must  R  be,  and  since  the  ratio  of  the 
sidereal  year  in  seconds  T  and  R  is  constant,  and  that  T  = 
— ^^^,  X  R,  it  follows  that  factoring  R  factors  T  also. 
Wherefore,  the  numerical  expression  of  the  mean  radius  of 
the  real  elliptical  orbit,  by  the  ratio  ^,  compels  the  expres- 
sion of  the  time  of  its  description  by  the  ratio  J.  Now,  con- 
ceiving resistance  to  the  revolution  of  a  planetary  mass  with- 
out volume,  at  the  distance  i  or  x,  eliminated  by  compres- 
sion of  the  sun's  volume,  it  would  revolve  in  x  seconds. 
The  proportion  x^  :  x^  :  :  T^  :  R^  is  consequently  explicit  in 
Kepler's  third  law. 

"Whence:  x  =  ~  =  8".8i  15507443 1 13. 

Also:    21-^:^  =z^^=  I. 

Multiplying  the  equation  by  ^,  ^  =  '^^  =  {^y  =(152.- 
9982253113687)^  =  23408.4569484283500. 

A  Dynamical  Resolution  of  the  Equatorial  Radius. 

The  universal  law  of  gravitation  prescribes  the  multipli- 
cation of  the  moon's  sidereal  period,  t,  by  the  square  root  of 
I  -hm,  the  sum  of  the  relative  masses  of  earth  and  moon,  the 


I  6  EXACT    ASTRONOMY. 

division  of  the  product  by  the  periodic  time  of  a  hypothetical 
satelHte,  conceived  to  revolve  around  the  earth  at  the  dis- 
tance I,  and  the  involution  of  the  quotient  to  the  fractional  in- 
dex I  for  the  moon's  mean  distance  from  the  center  of  her  mo- 
tion. Conceiving  resistance  eliminated  by  compression  of  the 
earth's  volume,  the  periodic  time  in  seconds,  of  such  satel- 
lite, is  the  product  of  three  factors,  to  wit :  the  terrestrial 
orbital  arc  x,  the  square  of  the  solar  day  in  hours,  d",  and 
the  reciprocal  of  ( i-f  m)^;  ~^:^^  =  "^^  =  _.i6o5g:.^i.±m|  ^5 

L  \  /    '     .\d--4-(i  +  m)J  xd-  5075.4532287233088  *      ^ 

0996460061535(1  +  mj',  involved  to  the  index  3,  equals 
60.0294004722072(1 -f-m)l  The  simple  ratio  of  the  dis- 
tances of  sun  and  moon:  23408.45694842835-^60.029400- 
4722072(1  4-m)"  =  3^9-94987o634909i^    All  that  preccdes  is  rigorously 

demonstrated  by  the  absolute  identity  of  the  sesquiplicate 
ratio  of  the  astronomical  periodic  times  of  earth  and  moon, 
and  the  simple  ratio  of  their  mean  distances:  |'-:-'li^^=: 
_IilL^==  7700.3979709322854-^(1 -fm)l  The  cube  root  of 
the  square  of  same  equals  389.9498706349084-^(1  +  m)''. 
The  difference,  0.0000000000007  x  60.0294  x  2092608 ^  x  12 
=  0.01  inch,  is  the  necessary  consequence  of  incommensura- 
bility. With  Herschel's  value  of  the  sidereal  year,  which  is 
0^3  greater,  the  above  discrepancy  of  o.oi  inch  is  increased 
^0  33-^3  miles.  Consequently,  no  variation  whatever  from 
the  values  employed  in  the  computation  is  possible  ;  which 
is  admittedly  the  highest  order  of  proof.  I  will  assume  sV 
for  the  moon's  mass,  and  then  demonstrate  that  it  cannot  be 
either  more  or  less  :  (i  -h  m)^  =  (  1.0125)'^=  1.0020725. 

Consequently,    the    arc    in    terms   of    radius    that    the 
hypothetical   satellite  (under  the  conditions  specified)  would 

describe  in  one  second  :  6.2831853071794x1.0020725 1=0.001240^21 1761. 

5075-4532287233088  ^   ^        ' 

An  arc  so  small  cannot  be  sensibly  different  from  its  tangent. 
Hence,  (i -t-curv.)^— 1^==  tan.^  =  arc^=  0.0000015388927S8- 
352. 


EXACT   ASTRONOMY.  IJ 

Completing  the  square  and  extracting  root :        ' 
Curv.'-|-2  curv. -I- 1  =  1. 000001538892788352. 
Curv.  +  I  —  1.0000007694462. 

Curvature  =  0.0000007694462  is  identical  with  the  fall 
of  a  weight  (all  resistance  being  eliminated)  in  one  second 
at  sea  level  on  the  equator  ;  because  its  deflection  in  equal 
times  from  the  tangent  to  its  path  is  the  same,  whether 
dropped  from  rest  or  projected  laterally  with  any  velocity. 
The  acceleration  of  gravity  at  London  has  been  determined 
with  the  required  accuracy  :  32.1928306  feet  per  second. 

Gravity  is  as  the  square  of  the  number  of  vibrations 
by  the  same  pendulum  In  equal  times. 

A  pendulum  that  beats  seconds  at  London  loses  136 
beats  in  24  hours  at  sea  level  on  the  equator.  On  the 
equator  gravity  is  diminished  jJt  part  of  itself  by  the 
momentum  of  the  earth's  rotation.  All  resistance  elimi- 
nated, the  fall  of  a  weight  in  i  second  :     3^-1928306^  /86264\2  ^. 

^  2  V86400' 

f9  =  i6.ioi4958oo5563feet. 

Length  of  the  equatorial  radius  :  16. 10 14958005563  -f- 
0.0000007694462  =  20926084  feet. 

The   Figure  of  the  Earth   Dynamically   Determined; 

Pendulum  observations  have  demonstrated  the  regular 
increase  of  gravity  with  increase  of  latitude  ;  equal  to  tIt 
part  of  equatorial  gravity  at  the  pole. 

The  distance  a  weight  would  there  fall  in  i  second : 
33^^918306   ^   ^86^^2  X  ^=  16.1284914950125  feet. 

The  earth's  oblateness,  concluded  from  the  geodeticaH 
measurement  of  meridional  arcs,  is  about  irU  part  of  the 
equatorial  radius.      I  will  now  demonstrate  that  the  terres- 
trial radii  are  in   the  inverse  duplicate  ratio  of  gravity  at 


EXACT    ASTRONOMY. 


their  intersections  with  sea  level,  computing  by  this  rule  : 
length  of  the  polar  radius :  (f|-;f^^^ff^)^  x  20926084  =  20- 
856091    feet. 


Oblateness  : 


69993  — 


20926084     398.974 

The  identity  of  results  demonstrates  the  rule,  which 
certainly,  simply  and  economically  determines  any  radius 
that  intersects  land.  The  tangential  force  being  as  the 
square  of  the  distance  from  the  axis  of  rotation,  is  as  the 
square  of  the  latitude  cosine.  Corrected  for  rotation,  the 
fall  of  a  weight  at  London  :  ^g^.J^Lzg)-^  ^  ^6.0964153  =  16.- 
1 1 801 25916476  feet  in  one  second. 

Radius:  (^ 6. 10x4958005563)2  ^  20,926,084  =  20,883,218  feet. 

\16.1180125916476  /  ' -/         7         -r  '         ,j' 

A  pendulum  that  beats  seconds  at  London  gains  79 
beats  in  24  hours  at  Spitzbergen,  in  latitude  79°  50'. 
Correcting:  for  rotation,  the  fall  of  a  weiefht  :    — ^ — ^  x 

'^  '  o  289-(o.i765i)' 

(11^)^  X   16.0964153==  16. 1 276029752894  feet  in  one  second. 

^^^^^^  •  (ll^f^SS^)'  ^  20926084  =  20858389  feet. 

The  correction  for  altitude  is  equally  certain  and  easy, 
and  I  feel  confident  that  a  weighing  apparatus  that  would 
accurately  determine  the  intensity  of  gravity  at  sea  is  not 
beyond  the  ingenuity  of  American  inventors. 

With  this  adjunct  to  the  pendulum,  the  exact  figure  of 
the  earth  would  be  quickly  determined. 

Distance,  Mass  and  Dimensions  of  the  Sun. 

Distance  between  the  centers  of  terrestrial  and  lunar 
revolution  :    ^3408.456948425x^0926084 ^^^^^^^  ^^^^  j^^H^^^ 

With  the  kilometer,  3280.8992  feet  for  unit=:  1493027- 
69.32  kilometers. 

Taking  the  earth's  weight  as  i,  it  follows  from  the  law 
of  gravitation,  that  the  mass  of  the  sun  is  the   inverse  du- 


EXACT    ASTRONOMV.  1 9 

plicate  ratio  of  the  computed  periodic  times  of  the  supposed 
earth  and  satellite,  both  revolving  at  the  distance  i  : 

(       xd^      \2   __         d*         __      33^776      _-     '»^od.O'^ 
Vx(i4-m)»^/  (i+m)i  1.0041493  '^^     ^     ^' 

The  sun  and  earth  both  revolve  in  a  year  about  their 
common  center  of  gravity;  distant  from  the  sun's  center  ; 
92774117-^-330405  =  280.8  miles. 

The  earth  and  moon  both  also  revolve  about  the  outer 
end  of  the  earth's  radius-vector  eccentric  to  her  center  : 
239230-7-80  =  2990  miles.  But  being  always  in  the  direc- 
tion of  the  moon  from  the  center,  its  mean  position,  with 
reference  to  the  sun's  distance,  is  at  the  center.  The  sun's 
mean  apparent  diameter,  I923".6,  as  observed  at  Greenwich 
Observatory  in  latitude  5  r29',  depends  on  the  observer's 
distance  from  the  sun's  center,  which  is.  less  by  the  cosine  of 
the  latitude  than  to  the  earth's  center. 

Wherefore,    the   actual    diameter   of    the    sun : 
'''"y'Jtfr''"'  X   I923".6  =  865,180  miles. 

206264    .806247  -'         «-»  v/' 

His  volume:    (3-^^)^  =  (109.15)^  =  1,300,383  earths. 

Distance  and  Diameter  of  the  Moon. 

60.0294004722  X  1.00553639x3963.2735  =  239230.11 
miles  is  her  mean  distance  from  the  center  of  her  motion. 
Since  her  mean  position  is  in  the  plane  of  the  equator,  her 
mean  distance  from  the  center  of  the  earth  is  2,990  miles 
greater.  By  reason  of  proximity,  her  mean  apparent  diame- 
ter, 1854'',  measured  at  Greenwich,  varies  slightly  with  the 
latitude  of  the  observer's  station.  Consequently,  her  mean 
distance  from  the  earth's  surface  is  the  square  root  of  the 
the  sum  of  the  squares  of  the  sine  of  the  observer's  latitude 
and  the  distance  from  its  intersection  with  the  plane  of  the 
equator  to  the  moon's  center.      Hence,  her  mean  distance 


20  EXACT    ASTRONOMY. 

(rpm  Greenwich  in  lat.  51°  29':    [(239230  -f  2990—  2468)* 
+  (3101)']^  =  239,782  miles. 
Her  actual  diameter : 

239782  X  i854''-^2o6264.''8  =  2i55. 2  miles. 


A  Crucial  Test 

Of  the  foregoing  determinations  is  applied  by  the  observed 
duration,  118  seconds,  of  the  total  phase  of  the  solar  eclipse 
of  Jan.  ist,  1889,  at  Willows,  California,  in  latitude  39^^ 
It  is  obvious  that  the  diameter  of  the  moon  and  that  of  her 
shadow  parallel  to  its  motion  at  intersection  with  the  earth 
are  sections  of  an  equilateral  triangle  standing  on  the  sun's  di- 
ameter. It  is  equally  obvious  that  the  duration  of  the  observ- 
er s  immersion  in  the  moon's  shadow,  when  she  is  on  his  me- 
ridian, is  the  diameter,  perpendicular  to  her  direction,  of  the 
shadow's  intersection  with  the  earth,  divided  by  the  rate  of 
its  motion.  Because  of  the  great  eccentricity  of  the  moon's 
orbit,  her  motion  is  far  from  uniform,  and  the  velocity  of  the 
earth's  rotation  is  also  greatest  at  the  equator.  But,  in  con- 
sequence of  the  divergence  from  parallelism  of  the  planes 
of  the  two  motions  towards  the  east,  and  of  the  earth's  con- 
vexity, the  velocity  of  the  shadow's  intersection  therewith 
is.  the  difference  of  the  moon's  mean  velocity  and  the  equa- 
torial velocity  of  the  earth's  rotation  : 

g3923o X 6..83X853  _  3963-711X1:^x853^  =0.34854!  mile  pcr  second. 

2360591^.5  86400J  »^  T      ^-r  i 

Diameter  of  the  shadow's  intersection  with  the  earth 
perpendicular  to  the  moon  : 

118  X  0.348541  =41.127838  miles. 

Taking  the  apparent  semi-diameters  of  the  sun  and 
moon,  computed  for  date  of  the  eclipse,  as  given  in  the 
Nautical  A htanaCy  the  observer's  distance  from  the  sun's 
center:    927741 17  x962^^-h976''.2  — 91,424,606  miles. 


OT  THB 


EXACT    ASTRON 

Now,  let  X  equal  the  leagth  of  segmen^^^^Prdae-^oon's 
shadow  cut  off  by  the  earth,  and  we  have  from  similar  tri- 
angles the  proportion  : 

X  :4i. 127838  :  :  x-h9i4246o6  :  865180. 
865 1 38.872 162X  =  3760096384.78 1828. 
x=: 4346. 23  miles. 
Also,    41.127838  :  4346.23  1:2155.2  :  moons    shadow. 
Total  length  of  shadows  227,753  miles. 
The  observer's  distance  from  the  moon  : 
227753—4346  =  223407  miles. 
Her  mean  distance  from  the  center  of  her  motion: 
[(223407  -f  cosine  39^^°,  305^)'  -  (•'^ine  391^°,  2521)^]'^  x 

^^2lif  =  2-^0,226  miles. 
927"        y^' 

The  deficiency  of  4  miles  may  be  credited  to  error  of 
observation. 

A  Sophistical  Assumption  Refuted. 

Professor  E.  S.  H olden,  of  the  Lick  Observatory,  and 
a  number  of  College  Professors  of  Mathematics  have  assert- 
ed that  my  resolution  of  the  solar  parallax  is  fallacious,  be- 
cause time  and  distance  cannot  be  measured  by  the  same 
unit;  and  that  the  expansion  or  contraction  of  the  earth 
would. not  alter  its  periodic  time.  The  sophistry  of  the  first 
objection  is  apparent,  since  every  astronomer  assumes  that 
the  only  possible  unit  of  the  earth's  mean  distance  is  an  arc 
of  a  great  circle  of  the  celestial  sphere,  which  must  become 
the  unit  of  the  time  in  which  the  whole  circumference  is  de- 
scribed, when  taken  as  the  unit  of  circumference.  The 
second  objection  is  fallacious,  puerile  and  absurd  because, 
although  true  of  abstract  time,  the  numerical  expression  of 
the  periodic  time  is  the  ratio  ot  the  periods  of  revolution  and 
rotation.  Thei^ubdi visions  of  the  latter  are  arbitrary  and 
their  number  constant.     By  the  laws  of  mechanics,  expan- 


22  EXACT    ASTRONOMY. 

sion  would  retard,  contraction  accelerate,  the  velocity  of  ro- 
tation. Consequently,  the  day  which  is  the  measure  of  the 
tangential  force  would  vary  by  the  expansion  or  contraction 
of  its  subdivisions,  as  the  square  root  of  the  equatorial  radi- 
us; and  the  sidereal  year  would  vary  in  the  same  proportion, 
inversely.  Hence,  the  cube  of  the  constant  R,  divided  by 
the  square  of  the  varying  T,  would  always  express  the  solar 
parallax, — their  duplicate  ratio  the  mean  orbital  radius,  with- 
out change  of  abstract  time  or  distance.  The  whole  as- 
sumption of  my  critics  Is  sophistical  and  devoid  of  honesty. 

The  Velocity  of  Light. 

The  progressive  motion  of  light  is  demonstrated  by  the 
annual  apparent  oscillation  of  stars  In  the  ecliptic,  which  be- 
comes an  oval  of  decreasing  eccentricity  with  Increase  of  the  ^ 
star's  declination.  The  center  of  this  apparent  motion  Is  the 
star's  real  place,  and  can  arise  only  from  the  projection 
on  the  vault  of  the  celestial  sphere  of  the  terrestlal  orbit 
seen  In  perspective  and,  consequently,  foreshortened  to  Its 
major  axis  by  the  apparent  motion  of  a  star  In  the  plane  of 
the  real  orbit  and  perpendicular  to  Its  major  axis.  All  other 
stars  In  the  ecliptic  trace,  of  course,  the  diameter  that  Is  per- 
pendicular to  each.  The  longest  diameter  of  the  oval  traced 
by  stars  having  latitude  Is,  of  course,  that  diameter  of  the 
orbit  parallel  thereto. 

The  assumption  of  the  text-books  that  all  the  stars  in 
the  plane  of  the  orbit  exhibit  the  maximum  amoimt  of  aber- 
ration, and  that  a  star  exactly  at  the  pole  would  apparently 
describe  a  circle,  Is  manifestly  erroneous.  The  phenomenon 
of  aberration  results  directly  from  the  earth's  orbital  velocity, 
being  a  sensible  fraction  of  the  velocity  of  light,  which  pro- 
jects the  orbit  on  the  sky,  at  a  distance  R  x  ~  from  the  eye,  . 
R  representing  the  semi-diameter,  V  and   v  the  velocities. 


EXACT    ASTRONOMY.  23 

Hence,  the  aberration  of  stars  in  opposition  when  the  earth 
is  at  equinox  and  solstice,  orives  an  independent  measure  of 
orbital  eccentricity  that  is,  perhaps,  more  reliable  than  the 
sun's  disc.  The  time  in  which  the  earth  describes  the  max- 
imum arc  of  aberration  or  reduced  semi-major  axis  and  light 
describes  the  real  semi-major  axis  : 

3i558i49\3  x  2o".445-^  1296000"  =  497\8444i546. 
The  velocity  of  light :  927741 16.75  "^  497-84441  546  = 
186352  miles  per  second  : 

.3408.4569484^5  X  ^sf^  =  299898.5  kilometers  per  second. 

497.84441546  3280.8982  :7:7    :7      u  1 

Charlatanry  Unmasked. 

About  three  years  ago  the  scientific  journals,  led  by 
Nature,  executed  a  grand  fanfare  in  glorification  of  the  al- 
leged independent  experimental  determination  of  the  ve- 
locity of  light  by  Professor  Simon  Newcomb,  at  the  U.  S. 
Naval  Observatory  and  by  Professor  Michelson,  at  Cleve- 
land. 

My  determination  of  the  solar  parallax  had  been  sub- 
mitted to  Professor  E.  S.  Holden  several  months  before. 

With  the  value  of  the  equatorial  radius  accepted  by 
American  astronomers  :  3962.720  miles  i^vide  Professor  C. 
A.  Young's  The  Sun),  and  mine  of  the  solar  parallax,  the 
velocity  of  light  becomes  : 

23408.45694B43835  X  =o^3^6  ^  299856.5  kilometers  per  second; 

497.84441546  3280.8992  :7  7    ^      vj  r  ' 

which  is  the  exact  mean  of  299860  and  299853,  the  deter- 
mination^ in  question. 

The  allowance  of  30  kilometers,  plus  or  minus,  con- 
ceded for  possible  error,  has  proved  inadequate  to  the  cor- 
rection of  the  actual  error  of  the  value  adopted  for  the  equa- 
torial radius.  The  combination  of  this  velocity  with  Nyren's 
determination  of  aberration  for  the  sun's  distance,  which  is 


2  4  EXACT    ASTRONOMV. 

expanded  200,000  miles  thereby,  is  a  very  clumsy  device 
for  masking  the  fraud. 

The  sums  expended  by  the  different  civilized  govern- 
ments in  futile  efforts  to  exactly  determine  the  sun's  distance 
by  triangulation  to  the  celestial  bodies  and  by  geodetical 
measurements,  aggregate  many  millions  of  dollars.  The 
problems  of  the  solar  parallax  and  of  the  dimensions  and 
figure  of  the  earth,  are  exhaustively  solved  by  these  few 
equations. 

I  have  now  arrived  at  the  conviction  that  the  solar  hori- 
zontal parallax,  as  defined  by  the  astronomers,  is  not  com- 
prehended by  the  average  reader. 

Its  dynamical  resolution  is  the  key  to  the  exact  deter- 
mination of  all  the  numerical  relations  of  the  solar  system 
and  is,  in  fact,  the  foundation-stone  of  the  entire  structure  of 
mathematical  astronomy.  The  name  discourages  the  non- 
mathematical  reader,  but  the  subject  itself  is  simple  and 
within  his  mental  reach,  when  accurately  described  in  simple 
terms. 

Although  the  earth's  orbit  is  not  exactly  a  circle,  it  is 
exactly  equal  to  the  circumference  of  a  circle  whose  radius 
equals  the  earth's  mean  distance  from  the  center  of  her  mo- 
tion. Now,  the  circumference  of  every  circle  contains  i,- 
296,000",  and  the  segment  thereof  which  is  equal  to  its  ra- 
dius contains  2o6264".8o6247 102477.  J- he  plane  of  the 
earth's  equator  is  always  nearly  coincident  with  the  plane  of 
her  orbit — exactly  so  when  she  is  at  the  vernal  or  autumnal 
equinox.  At  those  times  her  equatorial  radius  occupies  a 
small  segment  of  the  above  described  great  circle  that  con- 
tains an  unknown  number  x  of  seconds.  Said  arc  is  the 
thing  named  by  the  astronomers  solar  horizontal  parallax. 
It  is  now  obvious  that  the  ratio  ^°^^^4fs+  is  the  ratio  of 
the  mean  orbital  radius  and  equatorial  radius  of   the  earth. 


EXACT    ASTRONOMY.  25 

We  remain  ignorant  of  the  sun's  actual  distance  till  this  ratio 
is  multiplied  by  the  ascertained  length  of  the  earth's  equa- 
torial radius. 

The  so-called  wonderful  difficulty  and  complexity  of  the 
problem  of  the  sun's  distance  have  been  enlarged  upon  by 
all  the  astronomers,  but  I  have  shown  most  conclusively 
that  the  problem  is  an  extremely  simple  one  and  is  easily 
solved  by  elementary  mathematics. 

Since  the  astronomers  are  themselves  lost  in  the  fog 
raised  by  their  conception  of  solar  parallax,  which,  rightly  con- 
ceived, is  the  key  to  the  exact  determination  of  all  astro- 
nomical maofnitudes,  whether  of  distance,  mass  or  volume,  I 
will  endeavor  to  make  the  thing  so  styled  clear  to  the  minds 
of  my  non-mathematical  readers,  by  detaching  it  from  all 
ideas  of  parallax.  The  earth's  orbit  is  an  ellipse  of  small 
eccentricity,  but  is  exactly  equal  to  a  circle  the  radius  of 
which  is  the  earth's  distance  from  the  center  of  her  motion 
when  she  is  at  the  vernal  or  autumnal  equinox ;  she  is  then 
at  her  mean  distance. 

At  those  times,  the  plane  of  her  equator  exactly  coin- 
cides with  the  plane  of  her  orbit,  and  her  equatorial  semi- 
diameter  is  concentric  and  coterminous  with  a  very  small 
segment  of  the  aforesaid  great  circle,  since  so  small  an  arc 
is  .not  sensibly  different  from  its  chord.  This  minute  arc 
containing  the  unknown  number  x  seconds,  being  taken  as 
the  unit  of  the  radial  arc  is  made  thereby  to  measure  the 
whole  circumference,  and  therefore  to  measure  the  time  of 
its  description,  or  the  sidereal  year,  containing  3 1558 149.3 
time  seconds,  equal  to  129600c''  x  24.3504238425926.  It  is 
now  self-evident  that  these  arcs  of  the  said  great  circle  are 
in  the  same  ratio  as  the  earth's  mean-orbital  and  equatorial 
radii.  The  sesquiplicate  ratio  of  the  time  and  distance  units 
must  be  the  same  as  that  of  the  wholes  ;  whence,  and  from 


26  EXACT    ASTRONOMY. 

Kepler's  third  law  :  ^  =x  =4^  -  8".  8r  155074431 13, -|- = 
206264".  806247102477  —  2  ?4o8.4  ^6048428  3  S' 

8".8ii5507443"3  '^^  TJ      :7T     T         ^u 

Also.  |-=(-^)^  =  (i52.9982253ii3687)'  =  23408.4569- 

4842835. 

We  remain  icrnorant  of  the  sun's  actual  distance  till  this 
ratio  is  multiplied  by  the  ascertained  length  of  the  earth's 
equatorial  radius,  to  the  exact  determination  of  which  the 
arc  X  is  the  key. 

Parallax  is  a  distant  object's  apparent  displacement,  re- 
sulting from  the  observer's  translation  to  another  point  of 
view,  transverse  to  the  direction  of  the  object.  Applied  to 
the  arc  x,  the  term  is  fallacious  and  misleading,  because  im- 
plying its  resolution  by  triangulation,  which  is  the  essential 
principle  of  all  the  methods  hitherto  employed.  The  di- 
verse results  obtained  are  inevitably  vitiated  by  instrumen- 
tal and  atmospheric  instability. 

It  is  now  patent  that  the  annoying  complexity  of  the 
problem,  so  much  bemoaned  by  the  astronomers,  is  of  their 
own  weaving  and  entirely  foreign  to  its  wonderful  natural 
beauty  and  simplicity.  Finally,  astronomical  treatises  are 
devoted  ad  nauseam  to  glorification  of  the  science,  to  the 
apotheosis  of  its  founders,  and  to  mutual  back-scratching 
of  its   votaries. 

The  deduction  of  solar  parallax  and  of  the  earth's  di- 
mensions from  the  periodic  times  of  earth  and  moon,  and 
from  the  acceleration  of  terrestrial  gravity,  is  what  is  meant 
by  the  dynamical  resolution  of  the  problems,  and  is  wholly 
original  with  the  author.  These  data  are  positively  known  ; 
all  instrumental  determinations  being  admittedly  inaccurate. 


'V 


,:W 


